Optimal. Leaf size=81 \[ \frac {(1-2 x)^{5/2}}{42 (3 x+2)^2}-\frac {71 (1-2 x)^{3/2}}{126 (3 x+2)}-\frac {71}{63} \sqrt {1-2 x}+\frac {71 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}} \]
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Rubi [A] time = 0.02, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {78, 47, 50, 63, 206} \[ \frac {(1-2 x)^{5/2}}{42 (3 x+2)^2}-\frac {71 (1-2 x)^{3/2}}{126 (3 x+2)}-\frac {71}{63} \sqrt {1-2 x}+\frac {71 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^3} \, dx &=\frac {(1-2 x)^{5/2}}{42 (2+3 x)^2}+\frac {71}{42} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2} \, dx\\ &=\frac {(1-2 x)^{5/2}}{42 (2+3 x)^2}-\frac {71 (1-2 x)^{3/2}}{126 (2+3 x)}-\frac {71}{42} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=-\frac {71}{63} \sqrt {1-2 x}+\frac {(1-2 x)^{5/2}}{42 (2+3 x)^2}-\frac {71 (1-2 x)^{3/2}}{126 (2+3 x)}-\frac {71}{18} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {71}{63} \sqrt {1-2 x}+\frac {(1-2 x)^{5/2}}{42 (2+3 x)^2}-\frac {71 (1-2 x)^{3/2}}{126 (2+3 x)}+\frac {71}{18} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {71}{63} \sqrt {1-2 x}+\frac {(1-2 x)^{5/2}}{42 (2+3 x)^2}-\frac {71 (1-2 x)^{3/2}}{126 (2+3 x)}+\frac {71 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 48, normalized size = 0.59 \[ \frac {(1-2 x)^{5/2} \left (245-284 (3 x+2)^2 \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{10290 (3 x+2)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 75, normalized size = 0.93 \[ \frac {71 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (120 \, x^{2} + 235 \, x + 101\right )} \sqrt {-2 \, x + 1}}{378 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 77, normalized size = 0.95 \[ -\frac {71}{378} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {20}{27} \, \sqrt {-2 \, x + 1} + \frac {225 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 511 \, \sqrt {-2 \, x + 1}}{108 \, {\left (3 \, x + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.70 \[ \frac {71 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{189}-\frac {20 \sqrt {-2 x +1}}{27}-\frac {4 \left (-\frac {25 \left (-2 x +1\right )^{\frac {3}{2}}}{4}+\frac {511 \sqrt {-2 x +1}}{36}\right )}{3 \left (-6 x -4\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 83, normalized size = 1.02 \[ -\frac {71}{378} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {20}{27} \, \sqrt {-2 \, x + 1} + \frac {225 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 511 \, \sqrt {-2 \, x + 1}}{27 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 63, normalized size = 0.78 \[ \frac {71\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{189}-\frac {20\,\sqrt {1-2\,x}}{27}-\frac {\frac {511\,\sqrt {1-2\,x}}{243}-\frac {25\,{\left (1-2\,x\right )}^{3/2}}{27}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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